Ncert Exercises & Solutions

In figure a body A of mass m slides on a plane inclined at angle $θ_{1}$ to the horizontal and g is the coefficient of friction between A and the plane. A is connected by a light string passing over a frictionless pulley to another body B, also of mass m, sliding on a frictionless plane inclined at an angle $θ_{2}$ to the horizontal.

(a) A will never move up the plane
(b) A will just start moving up the plane when $μ = \frac{\sin θ_{2} - \sin θ_{1}}{\cos θ_{1}}$
(c) For A to move up the plane, $θ_{2}$ must always be greater than $θ_{1}$
(d) B will always slide down with a constant speed

Which of the following statement/s is/are true?

1. (b, c)
2. (c, d)
3. (a, c)
4. (a, d)

(b, c) Hint: Apply Newton's second law of motion.

Step 1: Find the coefficient of friction.

Let A moves up the plane frictional force on A will be downward as shown.

When A just starts moving up

\begin{array}{l} mg \sin θ_{1} + f = mg \sin θ_{2} \\ \Rightarrow mg \sin θ_{1} + μmg \cos θ_{1} = mg \sin θ_{2} \\ \Rightarrow μ = \frac{\sin θ_{2} - \sin θ_{1}}{\cos θ_{1}} \end{array}

Step 2: Find the friction force.
When A moves upwards

\begin{array}{r} f = mg \sin θ_{2} - mg \sin θ_{1} > 0 \end{array}

\Rightarrow \sin θ_{2} > \sin θ_{1} \Rightarrow θ_{2} > θ_{1}

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